Question

At a school with 100​ students, 33 were taking​ Arabic, 39 ​Bulgarian, and 40 Chinese. 14 students take only​ Arabic, 19 take only​ Bulgarian, and 21 take only Chinese. In​ addition, 13 are taking both Arabic and​ Bulgarian, some of whom also take Chinese. How many students are taking all three​ languages? None of these three​ languages?

Answer 1

6 students are taking all three

20 students are taking none.

Explanation:

Please check image attached.

From the total of 33 that are taking Arabic, 14 only take Arabic and 13 take Arabic and Bulgarian (b + d), so the students that take only Arabic and Chinese together is:

a = 33 - 14 - 13 = 6

From the total of 39 that are taking Bulgarian, 19 only take Bulgarian and 13 take Arabic and Bulgarian, so the students that take only Bulgarian and Chinese together is:

c = 39 - 19 - 13 = 7

Now, from the total of 40 that are taking Chinese, 21 only take Chinese, 6 take only Arabic and Chinese and 7 take only Bulgarian and Chinese, so the number of students that take all three languages is:

d = 40 - 21 - 6 - 7 = 6 students

The number of students that take any language is 14 + 19 + 21 + 6 + 7 + 13 = 80 students, so 100 - 80 = 20 students take none of the three languages